We present a novel numerical scheme to approximate the solution map s ↦ u ( s ) := 𝓛 – s f to fractional PDEs involving elliptic operators. Reinterpreting 𝓛 – s as an interpolation operator allows us to write u ( s ) as an integral including solutions to a parametrized family of local PDEs. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike prior works, we deduce the choice of snapshots for the reduced basis procedure analytically. The integral is interpreted in a spectral setting to evaluate the surrogate directly. Its computation boils down to a matrix approximation L of the operator whose inverse is projected to the s -independent reduced space, where explicit diagonalization is feasible. Exponential convergence rates are proven rigorously. A second algorithm is presented to avoid inversion of L . Instead, we directly project the matrix to the subspace, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.

中文翻译：

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分数扩散算子的简化基法 II

我们提出了一种新颖的数值方案来将解映射 s ↦ u ( s ) := 𝓛 – sf 近似为涉及椭圆算子的分数 PDE。将 𝓛 - s 重新解释为插值算子允许我们将 u ( s ) 写成一个积分，包括对参数化的局部 PDE 族的解。我们在有限元方法之上提出了一种简化的基础策略来逼近其被积函数。与以前的工作不同，我们通过分析推断了简化基础过程的快照选择。积分在光谱设置中被解释以直接评估代理。它的计算归结为算子的矩阵近似 L，其逆投影到与 s 无关的缩减空间，其中显式对角化是可行的。严格证明了指数收敛速度。提出了第二种算法以避免 L 的反转。相反，我们直接将矩阵投影到子空间，在其中评估其负分数幂。与前代产品的数值比较突出了其竞争性能。